Asymptotics for Szegö polynomial zeros

The Wiener-Levinson method and algorithm, formulated here in terms of Szegö polynomials _n ( _N,I ;z) orthogonal on the unit circle, is used to find unknown frequencies _j from anN-sample of a discrete time signal consisting of the superposition of sinusoidal waves with frequencies ₁,...,₁. In a recent paper the authors (and W.J. Thron) have shown that zerosz(j, n, N, I) of _n ( _N,I ;z) converge asN to the critical points ,j=1, 2,...,I, providednn ₀ (I)=2I+L, whereL is 0 or 1. The present paper gives results on the convergence of zerosz(j, n, N, I) to some of the for the case in whichnn ₀ (I), wheren is the degree of _n ( _N,I ;z).Research supported in part by the United States Educational Foundation in Norway (Fulbright Grant), the Norwegian Research Council (NAVF) and the US National Science Foundation under Grant No. DMS-9103141.     

The minimal polynomial of a matrix ∗

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A (λ) of A over F. By using q_A (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

Kato–Milne cohomology and polynomial forms

Given a prime number p, a field F with $\mathrm{char}(F)=p$ and a positive integer n, we study the class-preserving modifications of Kato–Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and p-regular forms. A p-regular form is defined to be a homogeneous polynomial form of degree p for which there is no nonzero point where all the order $p?1$ partial derivatives vanish simultaneously. We define a ${\stackrel{?}{C}}_{p,m}$ field to be a field over which every p-regular form of dimension greater than $p^m$ is isotropic. The main results are that for a ${\stackrel{?}{C}}_{p,m}$ field F, the symbol length of $H_p^2(F)$ is bounded from above by $p^{m?1}?1$ and for any $n??(m?1){\log }_2?(p)?+1$, $H_p^{n+1}(F)=0$.     

Does polynomial parallel volume imply convexity?

For a non-empty compact set A^d, d2, and r0, let Ar denote the set of points whose distance from A is r at the most. It is well-known that the volume, V_d(A_r), of A_r is a polynomial of degree d in the parameter r if A is convex. We pursue the reverse question and ask whether A is necessarily convex if V_d(A_r) is a polynomial in r. An affirmative answer is given in dimension d=2, counterexamples are provided for d3. A positive resolution of the question in all dimensions is obtained if the assumption of a polynomial parallel volume is strengthened to the validity of a (polynomial) local Steiner formula. Mathematics Subject Classification (2000):52A38, 28A75, 52A22, 53C65     

Representations of polynomial Rota–Baxter algebras

A Rota–Baxter operator is an algebraic abstraction of integration, which is the typical example of a weight zero Rota–Baxter operator. We show that studying the modules over the polynomial Rota–Baxter algebra $(\mathbf{k}[x],P)$ is equivalent to studying the modules over the Jordan plane, and we generalize the direct decomposability results for the $(\mathbf{k}[x],P)$-modules in [13] from algebraically closed fields of characteristic zero to fields of characteristic zero. Furthermore, we provide a classification of Rota–Baxter modules up to isomorphism based on indecomposable $\mathbf{k}[x]$-modules.     

Conditions for polynomial Liénard centers

In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coupled functional equations are solvable, in this paper we give an algorithm to compute a Gr¨obner basis for irreducible decomposition of algebraic varieties so as to find algebraic relations among coefficients of the damping function and the restoring function. We demonstrate the algorithm for polynomial Li′enard systems of degree 5, which are divided into 25 cases. We find all conditions of those coefficients for the polynomial Li′enard center in 13 cases and prove that the origin is not a center in the other 12 cases.     

Trigonometric polynomial or exponential fitting approach?

The application of a trigonometric polynomial and an exponential fitting approach is compared for a three-point formula for second-order derivatives, for Simpson’s quadrature rule and for Numerov’s scheme for second-order differential equations. The expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied. The errors for specific problems obtained in both the approaches as a function of the frequency are compared.     

Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G=G(t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual functionals for the G-Bernstein basis functions and we use this dual functional property to prove that G-Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G-Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G-Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G-Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G-Bézier curve, and we derive the geometric rate of convergence of this algorithm.     

Igusa’s p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure

For p prime, we give an explicit formula for Igusa’s local zeta function associated to a polynomial mapping ${{\bf f} = (f_1, \ldots, f_t) : {\bf Q}_p^{n} \to {\bf Q}_p^{t}}$ , with ${f_1, \ldots, f_t \in {\bf Z}_p[x_1, \ldots, x_n]}$ , and an integration measure on ${{\bf Z}_p^{n}}$ of the form ${|g(x)||dx|}$ , with g another polynomial in Z _p [x ₁, . . ., x _n ]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F _p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results in Denef and Hoornaert (J Number Theory 89(1):31–64, 2001), Howald et?al. (Proc Am Math Soc 135(11):3425–3433, 2007) and Veys and Zú?iga-Galindo (Trans Am Math Soc 360(4):2205–2227, 2008).     

Interpolating polynomial wavelets on [−1,1]

The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallée Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms. Dedicated to Prof. Guiseppe Mastroianni on the occasion of his 65th birthday.AMS subject classification 65D05, 65T60     

Some extremal properties of multivariate polynomial splines in the metric Lp(?d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (ℝ^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (ℝ^d in the metric L_p((ℝ^d).     

Bézoutians and injectivity of polynomial maps

We prove that an endomorphism f of affine space is injective on rational points if its Bézoutian is constant. Similarly, f is injective at a given rational point if its reduced Bézoutian is constant. We also show that if the Jacobian determinant of f is invertible, then f is injective at a given rational point if and only if its reduced Bézoutian is constant.     

The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

Kötter interpolation in skew polynomial rings

Skew polynomials are a noncommutative generalization of ordinary polynomials that, in recent years, have found applications in coding theory and cryptography. Viewed as functions, skew polynomials have a well-defined evaluation map; however, little is known about skew-polynomial interpolation. In this work, we apply Kötter’s interpolation framework to free modules over skew polynomial rings. As a special case, we introduce a simple interpolation algorithm akin to Newton interpolation for ordinary polynomials.     

Unconstrained and convex polynomial approximation inC[−1,1]

Some estimates for unconstrained and convex polynomial approximation in the uniform metric are obtained. These results are given in terms of the Ditzian-Totik moduli of smoothness , ≤1 with . The construction of the approximating polynomials does not depend on λ.     

Padé polynomial asymptotics from a singular integral equation

We derive a singular integral equation satisfied by the remainder function associated with the polynomials forming a diagonal Padé approximant. From this equation, the asymptotic behavior of the high-order polynomials is deduced for certain classes of functions being approximated.Communicated by Edward B. Saff.     

Small-amplitude limit cycles of polynomial Liénard systems

In this paper,we study the number of limit cycles appeared in Hopf bifurcations of a Linard system with multiple parameters.As an application to some polynomial Li’enard systems of the form x=y,y=gm(x)-fn(x)y,we obtain a new lower bound of maximal number of limit cycles which appear in Hopf bifurcation for arbitrary degrees m and n.     

A new central polynomial for 3 × 3 matrices

Abstract

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations.